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Equivariant compactifications of reductive groups
Dmitri A. Timashev Moscow State University


Abstract. We study equivariant projective compactifications of reductive groups obtained by closing the image of a group in the space of op erators of a projective representation. (This is a "non-commutative generalization" of projective toric varieties.) We describ e the structure and the mutual p osition of their orbits under the action of the doubled group by left/right multiplications, the lo cal structure in a neighb orho o d of a closed orbit, and obtain some conditions of normality and smo othness of a compactification. Our approach uses the theory of equivariant emb eddings of spherical homogeneous spaces and of reductive algebraic semigroups.

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1. Projective emb eddings of reductive groups Let G b e a connected reductive complex algebraic group. Examples. G = GLn(C), S Ln(C), S On(C), S pn(C), (C×)n. Let G P(V ) b e a faithful projective representation. It comes from a faithful rational linear representation G V , where G G is a finite cover. G End V = G P(End V ) Objective. Describ e X = G P(End V ). Example 1. G = T = (C×)n an algebraic torus; 1, . . . , m Zn the eigenweights of T = T V = X is a projective toric variety corresp onding to the p olytop e P = conv{1, . . . , m}.
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Problem 1. Describ e (G × G)-orbits in X : dimensions, representatives, stabilizers, partial order by inclusion of closures. Problem 2. Describ e the lo cal structure of X . Problem 3. Normality of X . Problem 4. Smo othness of X . Relevant research. 1) Affine emb eddings of reductive groups = reductive algebraic semigroups (Putcha-Renner, Vinb erg [Vi95], Rittatore [Ri98]). 2) Regular group compactifications: cohomology (De Concini- Pro cesi [CP86], Strickland [St91]), cellular decomp osition (Brion- Polo [BP00]). 3) Reductive varieties (Alexeev-Brion [AB04], [AB04']).
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Notation. T G (resp. T G) a maximal torus; weights T C× ,
l1 t t := t1 ž ž ž tln , n

Zn the weight lattice,

= ( l1 , . . . , l n ) , t = (t1, . . . , tn) T ;

(V ) = {eigenweights of T V }; P = conv (V ) the weight p olytop e; = G the set of ro ots (= nonzero T -eigenweights of Lie G), = + - (p ositive and negative ro ots); W = NG(T )/T the Weyl group; (ž, ž) a W -invariant inner pro duct on ; C = CG = { Z Q | (, ) 0, +} the p ositive Weyl chamb er. It is a fundamental domain for W Z Q.
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2. Orbits For each face F of P (of any dimension, including P itself ) let: VF V b e the span of T -eigenvectors with weights in F ; VF V b e the T -stable complement of VF ; V = VF VF ; EF = projector V VF . Theorem 1. There is a 1-1 corresp ondence: (G × G)-orbits Y X faces F P , (int F ) C = .

Orbit representatives are: Y y = EF . Stabilizers are computed. Dimensions: dim Y = dim F + \ F . Partial order: Y1 Y2 F1 F2.
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2 Example 2. G = P GLn, V = Cn = X = P(Matn) = Pn -1. Here G = S Ln, T = {diagonal matrices},

(V ) = {1, . . . , n} = the standard basis of Zn, = {i - j | i = j }, + = {i - j | i < j }, C = { = ( l 1 , . . . , l n ) | l1 ž ž ž l n } We see that P = conv{1, . . . , n} is a simplex, the faces of P whose interior intersects C are Fr = conv{1, . . . , r }, the resp ective projectors are EFr = diag(1, . . . , 1, 0, . . . , 0),
r

r = 1, . . . , n,

and the orbits are Yr = P(matrices of rank r).
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Pro of. Step 1. Let K G b e a maximal compact subgroup; then G = K T K (Cartan decomp osition) = X = K T K = T intersects all (G × G)-orbits in X . Step 2. By toric geometry, there is a 1-1 corresp ondence: T -orbits in T all faces F P ,

which resp ects partial order, EF b eing the orbit representatives. Step 3. Compute the stabilizers of EF in G × G. In particular, this yields the orbit dimensions. Step 4. Given y = EF Y = Gy G, the structure of (G × G)y implies that Y diag T = w1,w2W T w1y w2? It follows that Y1 = Y2 F1 = wF2 for some w W. There exists a unique F + = wF such that (int F +) C = .
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3. Lo cal structure Fix a closed orbit Y0 X . neighb orho o d of Y0? What is the structure of X in a

W.l.o.g. V is assumed to b e a multiplicity-free G-mo dule. By Theorem 1, Y0 = Gy0G, y0 = E0 , 0 P a vertex, E0 is the projector V = Cv0 V Cv0 , where v0 is the (unique) 0 eigenvector of weight 0 (a highest weight vector). Asso ciated parab olic subgroups: P + = G v , P - G. 0 + Levi decomp osition: P + = Pu L, L = P + P - T . œ Theorem 2. X = {x = A X | Av0 V } is a (P - × P +)/ 0 stable neighb orho o d of y0 in X . œ X
+ - Pu × Z × Pu ,

where Z = L End(V0 (-0)).
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Pro of is based on the lo cal structure of P - V in a neighb orho o d of v0 (Brion, Luna, Vust). Remark. Z is a reductive algebraic semigroup with 0 (corresp onding to y0), called the slice semigroup. Regular case: 0 int C = L = T = Z affine toric variety. Example 3. In the notation of Example 2, Y0 = P(matrices of rank 1), 1 + Pu = 0 . . . 0 1 0žžž 0 ,
- Pu =

0 = 1,

v0 = e1,

E

E

,

0ž ž ž 0 L= 0 . . . 0



GLn-1,

V0 = e2, . . . , en ,

Z = Matn-1 .
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4. Normality Do es X have normal singularities? It suffices to consider singularities in a neighb orho o d of a closed orbit Y0. By Theorem 2 it suffices to study the singularity of Z at 0. L is reductive = L-mo dules are completely reducible. Simple L-mo dules V = VL() highest weights CL, VL() VL(÷) + (V ), / VL( + ÷) ž ž ž VL( + ÷ - 1 - ž ž ž - ( +. L k ) ž ž ž
+ i L ).

Definition. Weights ÷1, . . . , ÷k L-generate a semigroup if = {÷ | VL(÷) VL(÷i1 ) ž ž ž VL(÷iN )}. Observation: "generate" (in a usual sense) = "L-generate"; = fails in general.
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Theorem 3. Let 0, 1 . . . , m b e the highest weights of the simple G-submo dules in V and 1, . . . , r b e the simple ro ots in + \ +. Put = (director cone of P C at 0) . Consider G L the following conditions: (1) X is normal along Y0; (2) T is normal at y0; (3) is L-generated by i - 0, -j . (4) is generated by - 0, (V ). Then (1) (3) = (2) (4). Remark. (3) i - 0, -j L-generate a saturated semigroup.

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Example 4. G = S p4, V {0 = 31, 1 = 22} (i simple ro ots). Here L 1 - 0, -1 L-generate = X non-normal along

2 = S 3C4S0 ( 2 C4), the highest weights 0 denote the fundamental weights, i the S L2 × C×, L = {+2}. {b old dots} (Clebsch-Gordan formula). Y0; b ecomes normal if we add 2 = 21.
s s s w w w w

? ? ? ? ? ? s e e e e e e

Ås r Å rr Å Å r Å rr ÅÅ r rw Å r sÅ s s s e T ? ? e e ? e se s s s s e e e s s s s s ew d ? d ? d ? ds ? d s s s s ? ? ? e ? es e ? s s s s r Å rr ÅÅ Å rr ÅÅ r rr Å rs Å Å

2

1

P

2

C 0

1

s s w w w w s rr Å rr ÅÅ Å r Å rr s s k s w w ÅÅ r rr ÅÅ r ÅÅ rr Å r ÅÅ rw Å E

1

2

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5. Smo othness Theorem 4. X is smo oth along Y0 (1)&(2)&(3): (1) L = GLn1 × ž ž ž × GLnp . (2) L V (-0) is p olynomial. (3) [L V (-0)] [GLni Cni ], i. Remark. (1), (2), (3) are reformulated in terms of (V ). Idea of the pro of. X is smo oth Matn1 × ž ž ž × Matnp . Example 5. G weights). a) i < m: V = b) i = m: V = X smo oth. Z is smo oth Z

= S O2m+1, V = VG(i) (i are the fundamental

C2m+1, L GLi × S O2m+1-2i = X singular. ž Cm over L spinor mo dule GLm = m
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i


6. "Small" compactifications Let G b e a simple Lie group. We take a closer lo ok at X = G P(End V ) for "small" V = VG(i) (i are the fundamental weights) or V = Lie G (adjoint representation). Results: 1) (G × G)-orbits, their dimensions, Hasse diagrams of partial order. 2) Non-normal: (S O2m+1, i), i < m; (S p2m, m); (G2, 2); (F4, i), i = 3, 4. 3) Smo oth: (S Ln, i), i = 1, n-1; (S Ln, Ad), n 3; (S O2m+1, m); (S p4, 1); (S p4, Ad); (G2, 1).

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References [AB04] V. Alexeev, M. Brion, Stable reductive varieties I: Affine varieties, Invent. Math. 157 (2004), no. 2, 227-274. [AB04'] V. Alexeev, M. Brion, Stable reductive varieties I I: Projective case, Adv. Math. 184 (2004), no. 2, 380-408. [BP00] M. Brion, P. Polo, Large Schub ert varieties, Representation Theory 4 (2000), 97-126. [CP86] C. de Concini, C. Pro cesi, Cohomology of compactifications of algebraic groups, Duke Math. J. 53 (1986), 585-594.
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[Ri98]

A. Rittatore, Algebraic monoids and group emb eddings, Transformation Groups 3 (1998), 375-396. E. S. Strickland, Computing the equivariant cohomology of group compactifications, Math. Ann. 291 (1991), no. 2, 275-280. D. A. Timashev, Equivariant compactifications of reductive groups, Sb ornik: Math. 194 (2003), no. 4, 589-616. E. B. Vinb erg, On reductive algebraic semigroups, Lie Groups and Lie Algebras: E. B. Dynkin Seminar (S. Gindikin, E. Vinb erg, eds.), AMS Transl. (2) 169 (1995), 145- 182.

[St91]

[Ti03]

[Vi95]